Preserving a qubit during state-destroying operations on an adjacent qubit at a few micrometers distance

Protecting qubits from accidental measurements is essential for controlled quantum operations, especially during state-destroying measurements or resets on adjacent qubits, in protocols like quantum error correction. Current methods to preserve atomic qubits against such disturbances waste coherence time, extra qubits, and introduce additional errors. We demonstrate the feasibility of in-situ state-reset and state-measurement of trapped ions, achieving >99.9% fidelity in preserving an ‘asset’ ion-qubit while a neighboring ‘process’ qubit is reset, and >99.6% preservation fidelity while applying a detection beam for 11 μs on the same neighbor at a distance of 6 μm. This is achieved through precise wavefront control of addressing optical beams and using a single ion as both a quantum sensor for optical aberrations and an intensity probe with >50 dB dynamic range. Our demonstrations advance quantum processors, enhancing speed and capabilities for tasks like quantum simulations of dissipation and measurement-driven phases, and implementing error correction.

Our apparatus (Supplementary Fig. 1) consists of 171 Yb + ions trapped in a four-rod paul trap with radial secular frequencies (ω x ,ω z ) of around 2π× 1.1 MHz and axial trap frequencies (ω y ) of 2π× 270 kHz.The ground state hyperfine levels S 1/2 |F = 0, m F = 0⟩ and S 1/2 |F = 1, m F = 0⟩ (separated by 12.642813 GHz) are assigned as the |↓⟩ and |↑⟩ of effective spin-1/2 particle, respectively.A magnetic field (B) perpendicular to the ion chain provides the quantization axis and a Zeeman splitting of ∆ zm = 2π × 3.25 MHz between S 1/2 |F = 1, m F = 0⟩ and S 1/2 |F = 1, m F = 1⟩ levels.Global Doppler cooling, state-detection, and optical pumping all derived from a laser source (369nm-laser-1) along with repump beams (935 nm) are illuminated onto the ions in the XY plane.A probe beam (along z) is illuminated onto the ions through an addressing system of effective numerical aperture(NA) of 0.16 (1).This probe beam is resonant to D transitions to perform siteselective state reset or measurement.We use another 369nm(369nm-laser-2) source for the probe beams whose frequency can be independently tuned (without affecting the global detection and cooling beams) to either D transitions.The fluorescence from the ions along z is collected through an in-house built objective onto a PMT(Hamamatsu: H10682-210) through a pellicle beam splitter(45:55) (Thorlabs: BP145B5).The optical access to the ion in our apparatus, perpendicular to the probe beam direction (z), is limited to an NA of less than 0.1.Due to this limitation of our ion trap apparatus, we are unable to perform high-fidelity state detection of the ions while they are being probed using the state-detection beam.
The state-dependent fluorescence transmitted after the trap is also monitored using a CMOS camera C2 (FLIR: Blackfly S BFS-PGE-04S2M) as shown in Supplementary Fig. 1.A microwave field drives the |↓⟩ to |↑⟩ transition.An acoustic-optic modulator (AOM1) in a double pass configuration, placed after the 369nm-laser-2, is used as a switch with precise timing and power control for the probe light.The light is then coupled to a PM fiber which is then expanded using a single lens(L1) and is polarizationcleaned using a polarizer.The light is sampled onto a photodiode (PD) that is used to stabilize the intensity fluctuations using PID feedback to the AOM.The polarization-cleaned and power-stabilized light from the PM fiber illuminates a Digital Micromirror device (DMD) (Visitech Luxbeam 4600 DLP) placed in the Fourier plane.A motorized λ/2 waveplate(WP1) is placed after the DMD to control the final polarization of the light.The DMD is programmed with an aberration-corrected amplitude hologram generated from an iterative Fourier transform algorithm (IFTA)(Supplementary Note 8) to produce a Gaussian beam of waist w=1.50 (5) µm in the ion plane (IP2).The negative first-order beam diffracted from the hologram on DMD is then relayed to the ion through the reflection of the pellicle.A flip mirror placed before the intermediate image plane IP1 is used to image the IP1 onto a camera C1 for initial characterization.Due to the limitations of our trap parameters, such as maximum electrode voltage, we could trap two 171 Yb + ions with an inter-ion spacing no smaller than 9 µm = 6w.
Despite designating the ground state hyperfine levels of 171 Yb + ions (S 1/2 |F = 0, m F = 0⟩ and S 1/2 |F = 1, m F = 0⟩) with |↓⟩ and |↑⟩ of an effective spin-1/2 particle, the measurement and reset processes involve additional states.Consequently, the ions may ultimately occupy the states S 1/2 |F = 1, m F = −1⟩ and S 1/2 |F = 1, m F = 1⟩ outside the Hilbert space of the qubit.To model the dynamics of the ion pertaining to this work, we account for eight levels in our Hilbert space, 4 for S 1/2 and 4 for P 1/2 (Supplementary Fig. 2

Supplementary Note 2. RAMSEY INTERFEROMETRY
To estimate the Fidelity F 1|2 (main text Eq. 1), we use a set of Ramsey measurements to characterize the P AQM caused by the probe beam parked at a distance d from the ion-1.(Main text Fig. 2).Each set of measurements is initialized by a sequence of Doppler cooling for 2.5 ms, optically pump(global) to |↓⟩ (|0⟩) for 20 µs.The probe light is illuminated for a time T between two microwave π/2 pulses (detuned from a transition |↓⟩ to |↑⟩ by ∆ µw = 2π × 10 kHz) for a duration of about 6 µs each.Here, ∆ µw is chosen such that the time periods of the Ramsey oscillations are much smaller than the characteristic decay times (T * 2 ), while ensuring that ∆ µw is smaller than the microwave-induced Rabi oscillation frequency of 100 kHz.A detection step follows where the ions are illuminated by a global detection beam for 1.5 ms, during which the state-dependent fluorescence from the ions is collected using a PMT.Each such experiment is repeated 200 times, and the PMT counts are averaged over.The averaged PMT counts are then normalized using measured counts from preparing |↓⟩ and |↑⟩ states.The normalized fluorescence(≈ P (|↑⟩)) oscillates at a frequency of 10 kHz as the time T is varied.We denote the contrast of these oscillations by R c (T ).To extract the characteristic decay time(T * 2 ) of the Ramsey contrast R c (T ) for a given configuration of d, these Ramsey measurements are done with varying T (main text Fig. 2a).Using the preliminary coarse estimate of Ramsey contrast, T * 2 is roughly estimated, and the time interval between 10µs and 2T * 2 is divided into five intervals, with each interval containing 21 data points in a span of 200 µs.
After the Ramsey measurements for these five intervals, the PMT counts are fit using the following function to extract the As a baseline measurement, we characterize the Ramsey measurements with no probe beam during the wait time (Supplementary Fig. 3) and estimate that the T * 2 is much larger than 200ms.This large T * 2 corresponds to an infidelity (1-F 1|2 ) < 3 × 10 −5 .

Supplementary Note 3. FIDELITY ESTIMATION
To quantify how well the quantum state of ion-1 is preserved after an operation on ion-2, we use the fidelity metric (F 1|2 ) [1] defined as where ρ(0) and ρ(t) denote density matrix operators of ion-1 (assuming unentangled with ion-2) before and after a state-reset or measurement operation (performed for time t) on ion-2, respectively.This metric yields a different value based on the initial state of ion-1, and using numerical simulations (Supplementary Note 5), we find that ρ(0) = |2⟩ ⟨2| represents the worst case scenario (Supplementary Fig. 4).By analytically solving the master equation of the system we find that the Ramsey fringe contrast R c (T ) could be used to estimate the worst-case fidelity of ion-1 after an operation ion-2 for a time (T) using To derive the above expression analytically, we assume that the intensity of probe light decohering the ion-1 is very weak, that it causes a low probability of accidental measurement P AQM << 1.In this limit, consider the density matrix of ion-1 in a reduced Hilbert space with only (|0⟩ , |1⟩ , |2⟩ , |3⟩) states.We model the action of probe light using the collapse operators where γ n is the rate of collapse.The collapse operators and their rates depend on the transition the probe is driving and its polarization.For example, in state detection, only the probe with π polarization causes the AQM of ion-1.In the limit P AQM << 1 with γ << 1 representing the rate of P AQM we use the the following collapse operators

Transition Polarization Collapse operators D
For ion-1 initialized in ρ(0) = |2⟩ ⟨2| state, the final state of the ion-1 after the AQM due to weak probe for a time t is calculated by analytically solving the Lindblad master equation Here, H atom in interaction picture is given by Here ∆ µw , ∆ zm denote the detuning of the microwave field and Zeeman splitting, respectively.From the solution, we find the fidelity F 1|2 to be Similarly, after the Ramsey experiment (Supplementary Note 2), the normalized is given as (S6) The Ramsey fringe contrast is then given by for a positive integer m and assuming γ << ∆ µw we get Combining Eq.S5 and S7 we get S3.Further, the Ramsey fringe decays exponentially with a characteristic time T * 2 which leads to Supplementary Note 4. CALIBRATIONS

A. FP aberration phase profile calibration
We characterize optical aberrations in the entire beam path in terms of a Fourier plane (FP) phase map.The optical aberrations till IP1 (Φ ) light of intensity I = 5 × 10 −5 I sat and polarization I π /I = 1 3 is applied on ion-1 for 11µs.For comparison, the infidelity (1 − F 1|2 )is shown(dotted line) from Ramsey interferometry(Supplementary Note 2) where a probe light of similar parameters as above is illuminated on ion-1 during the wait time.
are characterized using the camera C1 [2] as a sensor to measure the relative optical phase between two FP 'patches'(Supplementary Fig. 5a).The optical aberrations from IP1 to IP2 (Φ ab ) are measured using a single ion as a sensor (see main text methods)(Supplementary Fig. 5b).The phase profile Φ (0) ab is used to compensate for optical aberrations using an iterative Fourier transform algorithm (IFTA) [2] to create a diffraction-limited gaussian beam spot in IP2.

B. Fourier plane intensity profile calibration
The incident light on DMD from L1 is nonuniform and has a Gaussian intensity profile.Further, the pellicle beam splitter (PB) has an angular dependence on reflection.The effective intensity profile on the FP is measured using an ion in IP2 as a sensor.The ion is prepared in |2⟩ state, and the optical pumping light from DMD is used to pump to |0⟩ state for a fixed time.The value of the intensity of probe light reflected by a circular patch(30 pixels diameter) on DMD is inferred from the decrease in ion fluorescence.This measurement is repeated for different phase-corrected (Supplementary Note 4 A) patches on the DMD to construct an effective intensity profile.The intensity profile is then smoothened and interpolated, and a square root of the intensity profile is used as the amplitude profile of the incident electric field.This amplitude profile is further used as an input to IFTA hologram generation algorithm [2].

C. Relative Intensity calibration
The intensity of the probe light illuminating the ion through DMD is controlled by adjusting the RF power of the AOM1(Supplementary Fig. 1).The RF power vs. intensity of the light is calibrated using camera C1.To ensure accurate reporting, the linearity of C1's exposure time is confirmed over four orders of magnitude from 100 µs to 5 s (Supplemen- Supplementary Fig. 6.C1 camera linearity calibration.When a probe beam of a given intensity illuminates the camera, the data represents the counts from the camera, corrected for background, as the camera's exposure is increased.Since the camera does not have a large dynamic range, the intensity of the incident probe beam is varied by a known amount, followed by the measurements taken within the dynamic range of the camera.All the data is then stitched together to obtain the plot above.Error bars represent the standard error from 5 measurement repetitions. tary Fig. 6).The linearity of the RF source power setting and the RF power output is calibrated using a spectrum analyzer.The pellicle placed after the IP1 has a polarization-dependent transmission profile that is calibrated using camera C2 and compensated using the AOM.This calibration also gives a relative measurement connecting the attenuated and unattenuated probe beam intensity(main text Fig. 3b and Fig. 4b) through the camera's exposure time and pixel intensity (at a fixed gain).

D. Absolute intensity and polarization calibration
The intensity of the probe beam (calibrated using C1) on the ion is calibrated with respect to the saturation intensity (I sat ) of the ion.A series of optical pumping experiments are done with the probe using varying calibrated power and input polarization(varied using the λ/2 waveplate WP).These experiments are then fit using numerical simulations to extract the absolute intensity and polarization of the light illuminating the ion.

E. Probe beam position and size calibration
The position of the probe beam and its beam waist is calibrated by using a single ion as a sensor for the intensity.The ion is initialized in state |2⟩, and the probe beam (state-reset) illuminates the ion for a time smaller than the optical pumping time, followed by a state measurement.The dependence of ion fluorescence as a function of beam position is used to extract the relative beam position and the beam waist (Supplementary Fig. 7).Here the position of the probe beam is changed by programming the hologram on the Digital micromirror device (DMD) to generate a shifted Gaussian beam.This procedure is regularly done before every set of experiments to fix the slow drift of the relative position of the beam to the ion.We measured that the probe beam drifts by about 0.20(15) µm over the period of 15 min (a single Ramsey measurement).

F. Length scale calibration
The imaging system's effective focal length (≈ 24mm) translates the known length scale in the Fourier plane (FP) to the length scale in IP2.We find the relative beam positions of two ions in a trap using an experiment similar to (Supplementary Note 4 E).The inter-ion spacing could be calculated from the difference between the estimated relative beam positions of the two ions.This estimated inter-ion spacing is compared to an estimation of equilibrium positions (estimated from the measured trap frequencies) to further calibrate the system's effective focal length.We could calibrate the length scale in IP2 using this method to within 5% accuracy.

G. Frequency calibration
The relative shift of the laser frequency is calibrated by tuning the laser to the optical pumping transition and maximizing its pumping efficiency onto the ion.

Supplementary Note 5. SETUP FOR NUMERICAL SIMULATIONS OF LINDBLAD MASTER EQUATION
To model the dynamics of the ion pertaining to this work, the relevant levels are within the S 1/2 and the P 1/2 manifolds (Supplementary Fig. 2) The Hamiltonian, describing the interaction with radiation, accounts for couplings due to optical pumping, state detection, and the microwave.For the purpose of efficient numerical simulations, it is useful to remove the time dependence through a rotating transform (U (t)) [3] such that H rot = U HU † − U d dt U † .We find that when the optical pumping, detection, and microwave couplings are monochromatic, the solution of U exists, and we use this to remove the time dependence from our total Hamiltonian.With the time dependence of the Hamiltonian accounted for, the time evolution of the density matrix can be determined by solving the Lindblad master equation(Eq.S4) with appropriate collapse operators due to the spontaneous emission.Using such numerical simulations, the evolution of the density matrix is calculated in a Ramsey interferometry(Supplementary Note 2).T * 2 is extracted from the simulations, and the dependence as a function of input intensity on ion-1 is calculated.This dependence is used to extract the intensity crosstalk I X from measured T * 2 of Ramsey measurements.

A. Rabi fequencies
The rabi frequencies for a simple 2-level system are set according to the formula: where I is the intensity of the laser, I sat is the saturation intensity, and Γ is the spontaneous emission rate of the transition.In our case, we are interested in finding the rabi frequency pertaining to a specific transition i.e (2F ′ + 1)(2J + 1) J J ′ 1 F ′ F I The reduced matrix element between the J levels can simply be calculated from the decay rate of the excited state using Fermi's golden rule as follows: Since it is commonly used in 171 Yb + literature, we introduce the saturation intensity as defined for the 2 S 1/2 to 2 P 1/2 ignoring the internal structure: Combining these equations along with |E| = I/2cϵ 0 we get In the case of 171 Yb + , for all the allowed transitions between 2 S 1/2 and 2 P 1/2 , the second line of the above expression evaluates to 1/3, leaving us with a particularly simple expression for the rabi frequency

. 5 .
FP phase and amplitude profile.a)The aberration phase profile was measured using camera C1. b) The aberration phase profile is measured using the ion at IP2.For a-b, the piston and tilt terms are removed from the measured phase profiles, and the profile is further smoothened and interpolated.c) The scaled amplitude profile measured at IP2.The measured amplitude profile is smoothened, interpolated, and fit to 2D Gaussian.

.
Probe beam position and size calibration.The data represents state-detection fluorescence counts from a single ion after it is optically pumped using a state-reset probe beam with a position offset.Error bars indicate the standard error from 200 experimental repetitions.The dashed orange line represents the master equation simulation of a similar sequence, simulating the ion's state when a state-reset probe beam with a beam waist of w = 1.55 µm and a position offset of −0.32 µm is applied.